Integrated circuits (IC) typically contain millions of electronic elements within a single device. Parameter extraction, or simulation, of the electronic elements within an IC prior to the manufacture of such a device is essential. Simulation of each electronic element, and its interaction with its neighbors, allows computation of circuit element values before the IC is built. ICs operating at frequencies in the range of gigahertz, where internal structures are submicron in size and may operate near resonance, require accurate "full-wave" simulation.
As described in the parent application, U.S. patent application Ser. No. 08/904,488, incorporated herein by reference in its entirety, historically, IC elements were computed from the geometry of an element within an IC by using general purpose, computer based field solver tools based on finite-difference or finite-element schemes. Typical of these tools is a requirement for volume discretization. Discretization means that the full volume of an IC element is divided in many smaller "discrete" volumes and/or surfaces, yielding a large number of points. These large number of points are descriptive as a whole of the discretized element. In general, solutions for electric and magnetic fields are computed for each point. The number of points, or the level of accuracy, or both, are increased until the computation of electric and magnetic fields for each point reaches a desired level of accuracy. Using this discretization approach, as frequencies go up, the number of points required for a practical "full wave" simulation also goes up, resulting in large computation time and memory use for the simulation. Since many simulations need to be conducted for the design of an IC, it is desirable to perform an extraction, or simulation, in as short a time as possible.
The prior art used simulation tools based on layered media integral equation formulations using direct solutions (such as Hewlett Packard--Momentum and Sonnet).
These are 2.5D simulators, typically used by the microwave and antenna communities. However, since these tools employ direct solution methods, they are restricted to relatively small problems. In addition, the equations that they are based on become ill-conditioned at lower frequencies, resulting in numeric difficulties in computing the results.
Yet another approach of the prior art for performing simulations is the use of integral equation methods using iterative solutions based on the fast multipole method. An example of this approach is "FastCap: A multipole accelerated 3-D capacitance extraction program" IEEE Transaction on Computer Aided Design 10(10):1447-1459, November 1991, incorporated herein by reference in its entirety. In general, this type of integral equation schemes work by introducing additional equations to enforce boundary conditions at region interfaces. The introduction of multiple equations for multiple boundary conditions can result in a prohibitive increase in problem size, again presenting problems with computation time and memory usage.
Another approach to solve parameter extraction problems is the use of layered Green's functions. These functions have traditionally been used in a 2.5 dimensional (2.5D) simulation context where the radiating sources are essentially planar. i.e., being confined to infinitely thin sheets. In some applications. 2.5D modeling of the structures is adequate because conductor thickness generally is much smaller than conductor width. However, in IC and packaging contexts planar modeling is generally insufficiently accurate. Physical shrinkage of IC geometry size, which is approaching submicron dimensions, dictates that the thickness of conductors within an IC is often on the same order as the conductor width. This physical characteristic of internal IC structures reduces the applicability of a strictly planar oriented approach by introducing substantial errors.